I am looking for examples of stochastic differential equations $dX_t = b(t, X_t)dt + \sigma(t, X_t)dW_t$, that either have a unique strong solution, but multiple (in law) weak solutions, or which have no strong solutions, but again multiple (in law) weak solutions. Are there any, or there are results that prevent this from happening?
For the first question, I know that pathwise uniqueness implies uniqueness in law, but I don't see how strong uniqueness (pathwise uniqueness on an initial space with $\mathcal{F}_t$ being augmented natural filtration of $W_t$) implies pathwise uniqueness on any other space.
In addition, I would appreciate any examples, but ones where both different laws of solutions are not degenerate (i.e. $X_t \neq 0$) are surely more interesting.
UPD: It seems for the first case, we can consider an equation $dX_t = sgn(X_t)dW_t$, where $sgn(0) = 0$. Then the only strong solution is $X_t \equiv 0$, while another Brownian motion can be a weak solution (as in a famous Tanaka's example https://en.wikipedia.org/wiki/Tanaka_equation).